How to be successful as a research mathematician? Follow your gut

Is Maths Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths Eugenia Cheng Profile (2023)

For many people, the word mathematics evokes bad memories from their school years. Some develop an aversion that lasts a lifetime. Eugenia Cheng is on a mission to rectify that — to rid the world of what she calls ‘maths phobia’. Nature spoke to the mathematician, concert pianist, acclaimed author and passionate educator about her latest book, Is Maths Real?

Born in the United Kingdom into a family of Hong Kong origins, Cheng is scientist in residence at the School of the Art Institute of Chicago in Illinois, where she teaches maths to art students.

What drew you to mathematics research?

In school, I always wanted to know ‘why?’. Maths was the only subject that had full explanations. In every other subject, it seemed like you had to just believe somebody. There never seemed to be an explanation. There were laws. For example, you take measurements of a spring, draw a graph and derive Hooke’s law. But why is Hooke’s law true? You haven’t proved it, you’ve just demonstrated it.

Charting a course to make maths truly universal

It’s also the same reason that a lot of people are turned off maths. They see it as a bunch of rules that they have to follow, but they never know where those rules come from. If they ask, they might get told that it’s a stupid question. That’s what my latest book is about: all those really great questions, such as why –1 × –1 = 1.

And writing?

I find academic writing very austere — the sentences have too many subclauses and you almost have to draw a diagram to understand. When I’m writing, I imagine having a conversation. At school, my English teacher taught me how to write essays, and to start with structure. Because I type very fast — a very useful skill, which my mother encouraged me to acquire when I was 11 — I can more or less type at the same speed as I imagine having a conversation. I sometimes write 5,000 words a day. It soon scales up to a whole book.

In a similar way, I think that process is the content of pure maths. It’s not facts, it’s not outcomes. That is a big part of my message. It’s also about developing a way of deciding what we count as true. Pure maths uses logic, and everything has to be proved using just logic for it to count as true.

A lot of popular-maths books lure the reader in with mathematical games, but you don’t — why?

The more outreach I do, the more I suspect that my research is the opposite of recreational mathematics. My view is that maths is there to help us achieve things. I want to convince people that it can help them in their lives and to think more clearly about the world. It’s not recreational, but it is still a joyful experience, because shedding light on things is joyful.

In your book, you seem to apply your mathematical mind to everything. Is that so?

I just can’t help it, my brain really works that way. I used to think that maths was entirely logical and music was the extreme opposite: entirely emotional. Part of why I thought this was that so many people asked me whether the two were related that I rebelled. My teenage rebellions were mild, but that was one of them.

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But now I realize that, in maths research, you don’t just follow logical steps. If you do, you’ll never get anywhere interesting. You have to use your gut instinct and feel your way through something first, and then back it up with logic afterwards.

With music, it’s the converse. I use logical, analytical structures to understand the music in the first place. Western classical music is typically structured in sections, a bit like a book or film, with a beginning, middle and end. Chords, progressions and key changes also shape the piece. Once I’ve got all that, then I can let the emotions fly.

You also discuss how colonialist ideology has influenced modern mathematics?

Maths has been shaped by European academics for the past few hundred years — academics who decided that it needed a particular form of rigour to hold up to scrutiny. That programme was deliberately set up. It’s wonderful and very productive, and it enabled mathematics to reach a consensus about what counts as proof. But — and here is the big but — it’s all standing on developments from ancient cultures, including Babylonian, Mesopotamian, ancient Egyptian and those in the Arab world, India, China and so on. And, of course, it excluded women for ages.

But there is a lot of mathematical thinking in other parts of the world. So there’s a question of what we count as maths, and then there’s a meta-question: why should we, as part of the European framework, get to bestow the label of maths on things? If we don’t, then we’re keeping it out; but if we do, then it’s like saying: “Oh, well done.” We might marvel at, say, Amish people raising a barn or Inuit people building a kayak without any European formal maths training. We might declare that this is maths — and I think it is. But it’s problematic if we think we’re doing them some great honour by bestowing the label of maths on what they do.

I don’t have a solution. Sometimes I just get exhausted thinking my way through all these problems. But I think we should try to think through them.

This interview has been edited for length and clarity.

Competing Interests

The author declares no competing interests.